NOTES D ÉTUDES ET DE RECHERCHE

Transcription

1 NOTES D ÉTUDES ET DE RECHERCHE A TIME-VARYING NATURAL RATE OF INTEREST FOR THE EURO AREA Jean-Stéphane Mésonnier and Jean-Paul Renne September 2004 NER # 115 DIRECTION GÉNÉRALE DES ÉTUDES ET DES RELATIONS INTERNATIONALES

2 DIRECTION GÉNÉRALE DES ÉTUDES ET DES RELATIONS INTERNATIONALES DIRECTION DES ÉTUDES ÉCONOMIQUES ET DE LA RECHERCHE A TIME-VARYING NATURAL RATE OF INTEREST FOR THE EURO AREA Jean-Stéphane Mésonnier and Jean-Paul Renne September 2004 NER # 115 Les Notes d'études et de Recherche reflètent les idées personnelles de leurs auteurs et n'expriment pas nécessairement la position de la Banque de France. Ce document est disponible sur le site internet de la Banque de France « The Working Paper Series reflect the opinions of the authors and do not necessarily express the views of the Banque de France. This document is available on the Banque de France Website France.fr.

3 A Time-Varying Natural Rate of Interest for the Euro Area 1 J.-S. Mésonnier 2 Banque de France Monetary Policy Research Unit J.-P. Renne Banque de France Monetary Policy Research Unit September 15, We are grateful to Gilbert Cette, Laurent Clerc, Patrick Fève, Nicola Giammarioli, Enisse Kharroubi, Hervé Le Bihan, Julien Matheron, Adrien Verdelhan and participants in the Banque de France workshop on Small Monetary Macromodels (Paris, September 2004) and in the MMF Annual Conference 2004 (London, September 2004) for useful comments and suggestions. All errors remain our own. Opinions expressed re ect the authors view only and not necessarily the position of the Banque de France. 2 Banque de France, Monetary Policy Research Unit, SEPMF, Paris cedex 01. Corresponding author: J.S. Mésonnier, jean-stephane.mesonnier@banque-france.fr.

4 Résumé : Dans cet article, nous estimons, pour une zone euro synthétique sur la période 1979T1-2002T4, un taux d intérêt naturel variable dans le temps à partir d un petit modèle macroéconomique backward-looking. La méthodologie est proche de celle développée par Laubach et Williams (2003) pour les Etats-Unis. Le ltre de Kalman permet d estimer simultanément deux variables inobservables: l écart de production et le taux d intérêt naturel. Le modèle espace-état sous-jacent comprend une équation de demande agrégée et une courbe de Phillips. Nos hypothèses d identi cation utilisent une relation étroite entre les variations du taux d intérêt naturel et les uctuations de basse fréquence de la croissance potentielle. L écart de taux d intérêt, c est à dire la di érence entre le taux d intérêt réel et l estimation de son niveau naturel, constitue un outil intéressant pour l évaluation de la politique monétaire sur les deux dernières décennies. Alors que notre taux d intérêt naturel estimé semble assez robuste aux modi cations des spéci cations du modèle, la relative importance de l incertitude entourant les séries estimées entrave son intégration directe dans le processus d élaboration de la politique monétaire. Mots-clés : Taux d intérêt naturel, écart de taux d intérêt, politique monétaire, ltre de Kalman, écart de production. Abstract: In this article we estimate a time-varying natural rate of interest (TVNRI) for a synthetic euro area over the period 1979Q1-2002Q4 using a small backward-looking macroeconomic model, broadly following a methodology developed by Laubach and Williams (2003) for the United States. The Kalman lter simultaneously estimates two unobservable variables: the output gap and the natural rate of interest. The underlying state-space model incorporates an aggregate demand equation and a Phillips curve. Consistent with the theoretical intuition, our identifying assumptions include a close relationship between the TVNRI and the low-frequency uctuations of potential output growth. The resulting interest rate gap, that is, the di erence between the real rate of interest and its estimated natural level, provides us with a valuable tool for assessing the monetary policy stance in EU12 over the last two decades. While our TVNRI estimate seems quite robust to changes in model speci cations, the relatively high uncertainty surrounding the estimate hampers its direct integration into the policy-making process. Keywords: Natural rate of interest, interest rate gap, monetary policy, Kalman lter, output gap. JEL Classi cation: C32, E32, E43, E52. 2

5 Résumé non technique : Le concept de taux d intérêt naturel et son utilisation normative pour la politique monétaire sont généralement associés à l économiste suédois pré-keynésien Knut Wicksell (1898, 1906, 1907) selon lequel, there is a certain rate of interest on loans which is neutral in respect to commodity prices and tends neither to raise nor to lower them (1936, p. 102). D après Wicksell, stabiliser les prix consiste donc à aligner le taux d intérêt (réel) du crédit supposé égal au taux d escompte contrôlé par la banque centrale sur le taux d intérêt naturel qui varie en réponse aux chocs a ectant ses déterminants (réels), principalement la productivité du capital. Le regain d intérêt actuel pour le concept s inscrit dans le cadre néo-wicksellien d analyse de la politique monétaire défendu par Woodford (2003), dans lequel le taux d intérêt naturel prévalant dans la règle de Taylor varie continûment en réponse à divers chocs réels. L écart de taux d intérêt réel, c est à dire la di érence entre le taux d intérêt réel de court terme caractérisant la politique monétaire et son niveau d équilibre ou naturel, semble être un candidat intéressant pour l évaluation de la politique monétaire. Nous reprenons la méthodologie que Laubach et Williams (2003) ont développée pour les Etats-Unis et l appliquons à une zone euro synthétique sur la période 1979T1-2002T4. Trois points principaux distinguent notre approche de la leur. Premièrement, nous supposons que le processus inobservable décrivant les uctuations de basses fréquences communes au taux d intérêt naturel et à la croissance potentielle est stationnaire auto-regressif, fortement persistant mais non intégré d ordre un. Ceci nous permet d éviter la di cile réconciliation d un taux d intérêt d équilibre et d une croissance potentielle non-stationnaires avec la théorie économique. Deuxièmement, nous optons pour une dynamique du taux d intérêt naturel présentant un degré de sophistication intermédiaire entre celui de Laubach et Williams (2003) d une part et celui d Orphanides et Williams (2002) qui ne prennent pas en compte de possibles co-mouvements entre le taux d intérêt naturel et la croissance potentielle d autre part. En n, notre taux d intérêt réel est le taux d intérêt réel ex ante utilisant les anticipations d in ations issues du modèle lui-même et non celles issues d une modélisation externe et univariée de l in ation. L estimation du taux d intérêt naturel et l écart de taux qui en dérive constituent des outils intéressants pour l évaluation de la politique monétaire sur les deux dernières décennies. En particulier, la supériorité de telles estimations sur des ltres univariés classiques est illustrée. Cependant, l intervalle de con ance mesurant l incertitude inhérente au ltrage de Kalman est relativement importante. En outre, les erreurs de mesure en temps réel peuvent également être substantielles. 3

6 Non-technical summary: The concept of a natural real rate of interest and its prescriptive use for monetary policy is generally associated with the Swedish pre-keynesian economist Knut Wicksell (1898, 1906, 1907). According to this early contribution, there is a certain rate of interest on loans which is neutral in respect to commodity prices and tends neither to raise nor to lower them (1936, p. 102). In Wicksell s view, price stability thus depends on keeping the (real) interest rate of credit understood to be equal to the discount rate controlled by the central bank in line with the neutral rate of interest, which varies according to shocks a ecting its (real) determinants, mainly the productivity of capital. The recent revival of the concept owes to the Neo-wicksellian framework for monetary policy analysis advocated by Woodford (2003), where the neutral rate embedded in the Taylor rule varies continuously in response to various real disturbances. The interest rate gap, then de ned as the di erence between the real short term interest rate representative of monetary policy and its equilibrium or natural counterpart, seems to be an interesting candidate for assessing the current monetary policy stance. We apply the methodology rst developed by Laubach and Williams (2003) to a synthetic euro area over the period 1979Q1-2002Q4. However, our model speci cations depart from theirs in three signi cant ways. Firstly, we assume that the unobservable process that drives the low frequency common uctuations of both the NRI and potential output growth remains stationary autoregressive instead of nonstationary, although we expect it to be persistent. This allows us to avoid the di cult reconciliation of a nonstationary output growth and a nonstationary equilibrium real interest rate with both economic theory and intuition. Secondly, we opt for a speci cation of the NRI whose degree of sophistication can be viewed as a compromise between those of Laubach and Williams (2003) and Orphanides and Williams (2002), who do not assume any co-movements between the NRI and potential ouput growth. Lastly, we compute the real interest rate as a model-consistent ex ante real rate of interest, using the in ation expectations provided by the model instead of deriving them from univariate autoregressive models of in ation. The estimated NRI and the derived interest rate gap constitute valuable tools for assessing the monetary policy stance over the last two decades. In particular, the superiority of such estimates in comparison with classical univariate lters is illustrated. However, the con dence interval, integrating the uncertainty associated with Kalman ltering, remains relatively broad. Besides, the real-time misperception of the natural rate of interest can also be substantial. 4

7 1 Introduction In this paper, we estimate a time-varying natural rate of interest (TVNRI) for the euro area within the framework of a small backward-looking macroeconomic model and using the Kalman lter along the lines of Laubach and Williams (2003). The concept of a natural real rate of interest and its prescriptive use for monetary policy is generally associated with the Swedish pre-keynesian economist Knut Wicksell (1898, 1906, 1907). According to this early contribution, there is a certain rate of interest on loans which is neutral in respect to commodity prices and tends neither to raise nor to lower them (1936, p. 102). In Wicksell s view, price stability hence depends on keeping the (real) interest rate of credit understood to be equal to the discount rate controlled by the central bank in line with the neutral rate of interest, which varies according to shocks a ecting its (real) determinants, mainly the productivity of capital. Although one may consider that a natural rate of interest (NRI) appears implicitly as the intercept in popular interest rate rules of the kind rst proposed by Taylor (1993) roughly a decade ago, the recent revival of the concept owes much to the Neo-wicksellian framework for monetary policy analysis advocated by Woodford (2003), where the neutral rate embedded in the Taylor rule varies continuously in response to various real disturbances. The interest rate gap (IRG), then de ned as the di erence between the real short term interest rate representative of monetary policy and its equilibrium or natural counterpart, seems to be an interesting candidate for assessing the current monetary policy stance, notably as an alternative to measures that employ monetary aggregates or exchange rates. Hence, central banks and central bank economists have recently devoted much attention to these theoretical developments and the resulting empirical estimation strategies (see, e.g., ECB, 2004, Christensen, 2002, Williams, 2003, Crespo-Cuaresma et al., 2003, Basdevant et al., 2004). A more careful reading of this expanding literature however reveals two main approaches, depending on whether the focus is on short term or medium to long run implications of a non-zero gap and, simultaneously, on the degree of structure put into the models that yield the estimates. A rst strand of this new natural rate literature broadly follows the lines of Blinder (1998), Woodford (2003) or Neiss and Nelson (2001) and derives the natural rate of interest within the framework of detailed structural New Keynesian models (see, e.g., Giammarioli and Valla, 2003, Smets and Wouters, 2003, for applications to the euro area). From this perspective, the natural rate of interest equals the equilibrium real rate of return in an economy where prices are fully exible, or in other words, it is the real short term rate of interest that equates aggregate demand with potential output at all times. The emphasis is thus put on short term developments. Neiss and Nelson (2001) for instance develop a dynamic stochastic general equilibrium (DSGE) model with sticky prices that they calibrate to the UK economy: they then compute what Laubach and Williams (2003) term the higher frequency component of the natural rate 5

8 of interest and track the period-by-period movements in the real rate of interest that are required to keep in ation constant. As Larsen and McKeown (2004) state it, such a DSGE approach is a priori desirable, because it obviously enables a structural interpretation of the interest rate gap and its variations, which pure statistical approaches (such as a band pass or a HP lter) do not allow. However, as far as its prescriptive use for policy purposes is concerned, the advantage of such an approach over more statistical ones is not clear-cut, at least given that the calibration exercise of the derived models usually implies a signi cant amount of arbitrary assumptions. Besides, the natural rate of interest generated by a DSGE model appears in some cases to be substantially more volatile than the actual real rate, which makes policy use quite di cult (see, e.g., results in Smets and Wouters, 2003). Be that as it may, Neiss and Nelson (2001) for the UK as well as Giammarioli and Valla (2003) for the euro area provide promising results suggesting that their interest rate gap estimates have an informational content for in ation that could be used for policy purpose. Another strand of the literature follows Laubach and Williams (2003) and mixes the reference to simple macroeconomic models usually found in the monetary policy literature with the use of semi-structural methods such as the Kalman lter in order to estimate the natural rate of interest, the potential level of output and/or the natural rate of unemployment as unobserved variables (recent examples include Orphanides and Williams, 2002, Crespo-Cuaresma et al., 2003, Basdevant et al., 2004). In this view, the natural rate of interest is the real short term rate of interest consistent with output at its potential and in ation stable in the medium run, i.e. once the e ects of demand shocks on the output gap and supply shocks on in ation have completely vanished. Though it is less precise than the former, this latter de nition seems to be more tractable in practice and hence more widely accepted. In a very stimulating contribution, Orphanides and Williams (2002) warn against the adverse and often undervalued consequences of misperceptions in the true NRI and of its companion concepts of natural rate of unemployment and potential output in terms of the stabilisation properties of monetary policy rules that include such unobserved variables. They compute various statistical estimates of the natural rate including the output of simple state-space models of the kind introduced by Laubach and Williams and compare the properties of policies optimised so as to provide a good stabilisation performance of in ation and output, but which possibly underestimate the magnitude of mismeasurements in the natural rate of interest. They conclude that the costs associated with underestimating natural rate mismeasurement are signi cantly higher than those of overestimating it. It follows that, given the uncertainty, central bankers need to be extremely cautious regarding the policy implications of the interest rate gaps they compute. This lesson converges with the conclusion of Laubach and Williams, who point out the high uncertainty surrounding estimates of natural rates in general. Along with Larsen and McKeown (2004) however, we argue that there is a case for the use of natural rate estimates obtained via semi-structural techniques such as those employed by Laubach and Williams, 6

9 which strike a convenient compromise between the DSGE approach and purely statistical estimates such as the commonly used HP lter. When not used as a basis for real-time prescription nor as a rm anchor for monetary policy (as advocated e.g. by Christensen, 2002), such estimates of the natural rate and of the interest rate gap provide a useful tool for an ex post assessment of the policy stance. Moreover, and, by essence, since they allow for large changes in structural variables like the level of potential output and the NRI, they can deal with and reasonably account for the large shocks and many structural changes that have a ected European economies over the last two to three decades. Low-frequency movements of such variables remain a priori out of reach of a more structural approach like DSGE models, where aggregate relationships are expressed as log-linear approximations around a non stochastic steady state. We apply the methodology rst developed by Laubach and Williams (2003) to a synthetic euro area over the period 1979Q1-2002Q4. However, our model speci cations depart from theirs in three signi cant ways. Firstly, we assume that the unobservable process that drives, as in Laubach and Williams, the low frequency common uctuations of both the NRI and potential output growth remains stationary autoregressive instead of nonstationary, although we expect it to be persistent 1. This allows us to avoid the di cult reconciliation of a nonstationary output growth and a nonstationary equilibrium real interest rate with both economic theory and intuition. Secondly, as we express some doubt regarding the feasability of estimating our already unobservable NRI as the sum of two equally unobservable components, we opt for a speci cation of the NRI whose degree of sophistication can be viewed as a compromise between those of Laubach and Williams and Orphanides and Williams, who do not assume any co-movements between the NRI and potential ouput growth. Lastly, we compute the real interest rate as a model-consistent ex ante real rate of interest, using the in ation expectations provided by the model instead of deriving them from univariate autoregressive models of in ation as Laubach and Williams and others do. The maximum likelihood estimation involves the calibration of two ratios, and the choice of the calibrated ratios relies on several statistical criteria. Our TVNRI nevertheless appears to be robust to changes in both ratios. The estimated NRI and the derived interest rate gap the di erence between the real short term interest rate and the natural rate constitute valuable tools for assessing the monetary policy stance in EU12 over the last two decades. In particular, the superiority of such estimates in comparison with classical univariate lters is illustrated. However, the con dence interval, integrating the uncertainty associated with Kalman ltering, remains relatively broad. Besides, the real-time misperception of the 1 The speci cation of a nonstationary process for the natural rate of interest and/or the rate of growth of potential output is relatively common in the literature : see e.g. Laubach and Williams (2003), Orphanides and Williams (2002), Larsen and McKeown (2002), Fabiani and Mestre (2001) or Crespo Cuaresma and Gnan (2003). The random walk assumption for the natural rate of interest has the technical advantage of combining persistent changes in the unobservable component with a smooth accomodation of plausible but unspeci ed structural breaks in the e ective interest rate series over a period of estimation that generally covers the last two to three decades. Nevertheless, using a unobservable components setting for the euro area, Gerlach and Smets (1999) assume that potential output is I(1). 7

10 natural rate of interest (that can be approximated by the di erence between the two-sided estimates using the whole sample information and the one-sided estimates using only information up to time t) can also be substantial 2. The rest of the paper is organised as follows. Section 2 presents the data set. Section 3 introduces the model. Section 4 develops estimation issues. Lastly, Section 5 analyses the results and examines our estimated TVNRI. 2 Data The euro area time series are taken from ECB s AWM database and cover the period 1979Q1 2002Q4 with quarterly frequency (see Fagan et al., 2000). The rst year corresponds to the EMS entering into force. Whereas the real GDP gures provided by the AWM database are already seasonally adjusted, the HICP series is not and we hence adjust it using the Tramo/Seats procedure. We denote by y t the log real GDP. In ation is de ned as the annualised quarterly growth rate of the HICP series (in logs) and is denoted by t. The ex ante real short term rate of interest r t is obtained by deducting from the current level of the 3-month nominal rate of interest i t the one-quarter-ahead expectation of (quarterly annualised) in ation as derived from the entire model estimation (denoted with t+1jt below). A novelty of our approach is hence to compute an ex ante real rate of interest using modelconsistent in ation expectations instead of proxies for expectations as derived from univariate models of price dynamics or other external modelisation of in ation. Appendix 2 provides with an assessment of the quality of our model-consistent in ation expectations and compares them with alternative in ation expectations derived from both continuously updated univariate autoregressive models of in ation and the univariate time-varying coe cients procedure described by Stock and Watson (1996). Figure 1 plots our model-consistent in ation expectations together with the one-quarter-lead of quarterly annualised in ation. To end with, two variables are unobservable and constitute the state variables in the state-space model described in the following section, namely the output gap z t and the natural rate of interest rt. 3 Speci cations Our speci cations are close to these of Laubach and Williams (2003), themselves partly following the lines of Rudebusch and Svensson (1998). The model relies on six backward-looking linear equations. 2 Note that this measure of misperception is potentially optimistic since even in the one-sided case, the whole sample information has been used to estimate the model parameters and is consequently not exactly a real-time estimate of the NRI. 8

11 Figure 1: In ation, model-consistent in ation expectations and current annual increase in consumer prices. This backward-looking nature of the model makes it subject to the Lucas critique, according to which reduced-form relations in traditional macroeconomic models depend implicitly on the agents expectations of the policy process and are hence unlikely to remain stable as policymakers changed their rules. However, empirical backward-looking models without explicit expectations are still widely used for monetary policy analysis, as in Rudebusch and Svensson (1998, 2002), Onatski and Stock (2002), Smets (1998), Dennis (2001), Laubach and Williams (2003), Fagan, Henry and Mestre (2001) and Fabiani and Mestre (2004). Moreover, several articles suggest that such models appear to be fairly robust empirically, notably Rudebusch and Svensson (1998), Bernanke and Mihov (1998), Estrella and Fuhrer (1999), Dennis (2001) and Leeper and Zha (2002). The model consists of the following equations: 9

12 t+1 = A(L) t + B(L)z t + " t+1 (1) z t+1 = (L)z t + (L)(i t t+1jt rt ) + " z t+1 (2) rt = r + r a t (3) yt = y + y a t + " y t (4) a t+1 = a t + " a t+1 (5) y t = yt + z t (6) where L denotes the lag operator. We assume that the four shocks are independently normally distributed, their variance covariance matrix " is given by: 2 " = z y 0 0 : : : 0 2 a 3 ; 2 a = 1 (7) 7 5 The rst equation can be interpreted as an aggregate supply equation, or Phillips curve. It relates consumer price in ation to its own lags and to the lagged output gap. The second one is a reduced form of an aggregate demand equation, or IS curve, relating the output gap to its own lags and to the interest rate gap i.e. the di erence between the short term real rate and the natural rate of interest. Policymakers then control the in ation rate with a lag of two periods. The natural rate of interest is identi ed through the interest rate gap. More precisely, the output gap is assumed to converge to zero in the absence of demand shocks and if the real rate gap closes. In this model, stable in ation is consistent with both null output and interest rate gaps. Hence, our NRI could also be conveniently labelled as a nonaccelerating-in ation rate of interest (NAIRI). An important feature of the model is the fact that monetary policy only a ects the rate of in ation indirectly, via the output gap. Lastly, we take the nominal short rate of interest as exogenous, or, put di erently, the reaction function of the central bank remains implicit. Departing from common speci cations in the literature 3, we assume that the natural rate of interest rt follows an autoregressive process instead of a random walk, as speci ed by (3) and (5). The complete estimation of the model con rms that this process is in fact highly persistent (see the estimator of in Table 1), which ts our purpose of capturing large and low frequency uctuations in the level of the equilibrium real rate, as would the hypothesis of a nonstationary NRI also do. 3 See e.g. Laubach and Williams (2003), Orphanides and Williams (2002), or Larsen and McKeown (2004). 10

13 Nevertheless, assuming that the NRI follows a nonstationary process hinders the economic interpretation of the model, in particular if we assume, as suggested by economic theory, that potential growth yt shares common uctuations with rt 4. Indeed, the economic intuition underlying our speci cation choice in equations (3) to (5) refers to a basic optimal growth set-up (the textbook Ramsey model), where intertemporal utility maximisation by the representative household yields the following log-linear relationship between the real interest rate r and the (usually constant) rate of labor-augmenting technological change g, which is also the rate of growth of per capita output along a balanced-growth path 5 : r = g + (8) Assuming then that this trend growth rate g is in fact subject to low frequency uctuations, we get the intuition underlying our speci cation choice, where g is equivalent to our y a t. The (highly) autoregressive process denoted by a t aims hence at capturing low-frequency variations in potential output growth, under the assumption that these variations are common with those of the NRI. In addition to this persistent but stationary process, potential output growth consists in our model of another stationary component, which may account for other sources of discrepancies with the natural rate of interest e.g. due to shocks to preferences or changes in scal policies. Estimations show that a simple white noise is su cient to model this stationary component 6. These speci cations assume that potential GDP is an I(1) process, as is usual for the euro area 7. Our speci cations attempt to model the links between potential output growth and the natural rate of interest. In this respect, our approach lies between these of Laubach and Williams (2003) and of Orphanides and Williams (2002). In the former, rt is the sum of the trend growth rate, which also drives the low-frequency uctuations of potential output growth, and of a second speci c (possibly nonstationary) component. In the latter study, the natural rate of interest and potential output growth are completely 4 A nonstationary speci cation for the NRI and then potential output growth through the assumption of a randomwalk for a t would indeed imply that potential output is integrated of order two, which would be at odds with available evidence for the euro area. Besides, when translated into the set-up of a standard optimal growth model, this would mean a nonstationary path for the ratio of output to the stock of capital. 5 This relationship relies notably on the assumption of a standard utility function of the representative household u(c t) = Ct 1 =(1 ) with constant relative risk aversion (which corresponds to the inverse of the intertemporal elasticity of substitution) and where stands for the rate of time preference of consumers. 6 AR speci cations systematically lead to non-signi cant autoregressive coe cients. 7 See for example Gerlach and Smets (1998). Besides, both the ADF and Phillips-Perron tests clearly reject the null hypothesis of an I(2) log real GDP. 11

14 uncorrelated. This last assumption appears at odds with theoretical intuition and may potentially results in non-optimal exploitation of the data. Laubach and Williams approach features a higher level of complexity and therefore appears more attractive. However, this complexity raises numerous estimation problems, notably because it means extracting two unobservable components out of an already unobservable variable (rt ) which in turn is identi ed through the dynamics of an other unobservable variable (z t ) 8. Since we aim precisely at estimating a TVNRI in a way as transparent and robust as possible, we prefer to consider a single-process-driven NRI, yet presenting common uctuations with potential output growth. 4 Estimation The previous equations can be written in the state-space form, and the parameters can be estimated by maximisation of the likelihood function provided by the Kalman lter (see Annex D for the statespace representation of the present model). This lter is a recursive algorithm for sequentially updating a linear projection for a dynamic system. Given a set of measurement and transition equations, the Kalman lter provides the best linear unbiased estimator of the state variables ( ltered or smoothed 9 ) and a particularly attractive feature of this approach is its ability to quantify uncertainty around the estimated state variables 10. Estimation by the Kalman recursive equations requires the setting of initial values for the state vector, which comprises two blocks relative to a t (and its rst lag) on the one hand and to the output gap z t (and its rst lag) on the other. Natural candidates for the initial conditions are the unconditional mean and autocovariances of the unobservable variables in each block. Since we have not speci ed any equation for the dynamics of the nominal interest rate (in other words the reaction function of the central bank 8 In particular, some estimates of key-parameters when several components enter the dynamics of the NRI are very sensitive to the initial state values and variances. More precisely, if the NRI is assumed to follow a two-component process (rt = rat + t ) and if the variance re ecting the con dence on the initial value 0 is not large enough, the parameter r might be bounded to estimate the initial level of the NRI instead of assessing the extent to which the NRI and potential output growth uctuate together. 9 A ltered estimate is one-sided that is, it uses information only up to time t. A smoothed estimate is two-sided and uses information from the whole sample, up to time T. In this respect, the HP lter is a smoother, since it can be thought of as a two-sided moving average. 10 As pointed out by Hamilton (1986), two forms of uncertainties are associated with the estimated state vector of a statespace model. The rst one, the lter uncertainty, re ects the fact that the estimated state vector represents conditional expectations of true unobserved values. This rst uncertainty is due to Kalman lter estimation and would be present even if the true value of the model parameter were known. The second one, the parameter uncertainty re ects the uncertainty around the estimated parameters. 12

15 remains implicit), the derivation of the unconditional mean and variance of the output gap z t remains however out of reach. We hence resort to a common practice which consists in adopting relatively di use priors and assuming su ciently large values for the unobserved variance matrix (which measures the con dence in the priors). More precisely, we use the HP lter to get a prior estimate of the output gap. The ltered series is then used to get initial values for z t as well as to derive the output gap block of the covariance matrix of the initial state vector. By contrast, it is straightforward from equation (5) to derive the unconditional mean and autocovariances of the AR process a t as a function of the parameters. The maximisation of the log-likelihood computed by the Kalman lter then yields simultaneously the vector of parameters and the initial conditions for a t. Since a direct estimation by maximum likelihood provides unconsistent results, we resort to the calibration of two ratios of parameters, namely y = z (denoted with 1 ) and r = y (denoted with 2 ). As illustrated in Table 1, no consensus on the 1 ratio can be reached among estimation results for similar models of the US and EU economies 11. On the one hand and due to our speci cations this ratio should not be too small because this would amount to impose the strong hypothesis that the natural rate of interest and potential output growth have exactly the same uctuations. Indeed, potential output growth would then reduce to y a t, which turns out to be equal to y = r rt. On the other hand, too large a ratio entails a potential output growth very close to the observed GDP growth. Besides, the larger 1, the more volatile is potential output growth and the less volatile the output gap. To the extent that we prefer the output gap to be more volatile than potential output growth, we should choose a value for 1 below unity. Without constraining the ratio r = y, y is spontaneously estimated to be zero, suggesting that there is no common trend between the natural rate of interest and potential output growth. However, Figure 2 plots the natural rate of interest resulting from such an estimation together with the HP- ltered output growth and shows that the series exhibit marked common uctuations 12. It can consequently be assumed that information contained in the data is not optimally extracted by direct MLE. For this reason, we resort to a calibration of the ratio r = y. Available empirical evidence suggests that the natural rate of interest varies from one-for-one to ve-for-one with changes in the 11 In Table 1, estimates of y= z ranges from approximately 0 to The correlation coe cient of the two series is

16 Table.1 Estimation results from various studies Auth. area period z y P i MR EA 1979:1-2002: :06 2 lags 0.19 FM EA 1970:1-1999: / 0.12 PS GS GS GS GS EU 1975:1-1997: (5) EU 1975:1-1997: (5) EU (5=10) 1975:1-1997: EU 1990:1-1997: (5) EU (10) 1990:1-1997: PS U S 1975:1-1997: RS U S 1961:1-1996: S U S 1962:1-1997: S U S 1980:1-1997: LW U S 1961:1-2000: Table 1: MR : Mésonnier and Renne, this paper / PS : Peersman and Smets (1999) / GS : Gerlach and Smets (1999) / S : Smets (2000) / RS : Rudebusch and Svensson (1998) / LW : Laubach and Williams (2001) / FM : Fabiani and Mestre (2004, model 1) trend growth rate 13. As a result, we consider such an interval for the ratio r = y in the following Estimation results As regards the Phillips curve, the choice of the order of the lag-polynomial A(L) is based on the signi cance of the last lag included. Moreover, the null hypothesis that the coe cients of the in ation lags sum to one is not rejected by the data, leading to an accelerationist form of the Phillips curve. In other words, 13 Intertemporal elasticities of substitution (IES) estimated by Hall (1988) are small and not statistically di erent from zero (corresponding to in nite risk aversion). However, Ogaki and Reinhart (1998) argue that Hall s model is misspeci ed because the intratemporal substitution between nondurable consumption goods and durable consumption goods is ignored and in two contributions (1998a, 1998b), they obtain IES estimates between 0.27 and 0.77 (corresponding to risk aversions between 1.3 and 3.7). Barsky et al. use micro-data and estimate an IES of 0.18, implying a coe cient of risk aversion of Since we use quarterly growth rate of GDP, a relative risk aversion coe cient of 1 corresponds here to a value of 4 for the ratio r= y: That is why the interval considered in the following for this ratio is [4; 20]. 14

17 Figure 2: HP- ltered output growth & estimate of the natural rate of interest when the ratio r = y is not constrained (the HP- ltered output growth series is demeaned and rescaled). in ation depends only on nominal factors in the long run. As regards the IS equation, only the rst lag of the output gap is included 15. In addition, following Laubach and Williams, two lags of the rate gap enter this equation. However, since the estimation of two distinct coe cients for each of these two lags results in some unsatisfying compensation phenomena, we constrain the two coe cients to be equal. The model can then be rewritten: t+1 = 1 t + 2 t t 2 + z t + " t+1 (9) z t+1 = 1 z t + 2 z t 1 + (1 + L)(i t t+1jt rt ) + " z t+1 (10) rt = r + r a t (11) yt = y + y a t + " y t (12) a t+1 = a t + " a t+1 (13) y t = yt + z t (14) The numerical BFGS algorithm provided by GAUSS is applied to get the MLE 16, under xed 1 = 15 Including a second lag yields a small and non-signi cant second autoregressive parameter. 16 As regards the initialization of the optimisation algorithm, many starting values have been tested: our estimates appear 15

18 y = z and 2 = r = y. The choice of these ratios is based on three criteria: - rst, the Lagrange Multiplier test, whose main advantage is that the unrestricted MLE does not need to be known, is applied for many values of the ratios (f 1 ; 2 g 2 [0; 3] [4; 20]) and combinations leading to a p-value lower than 25% are rejected; - the economic relevance of the estimated unobservable components and the signi cativity of the main parameters of our model constitutes a second criterion; - the level of implied uncertainty inherent to the Kalman ltering procedure is the third one. Results according to the rst criterion 17 are presented on Figure 10, and suggest that 1 should be chosen lower than 2 and 2 greater than 8. Table 2 contains the parameter estimates when 1 is equal to 0.5 and 2 to di erent values (12, 16, 20 or 1); and when 2 is equal to 16 and 1 is equal to p 0:1 or 1:5. The computation of the information matrix is based on the expression given by Engle and Watson (1981) (see Annex C). All the parameters have the expected sign. The monetary policy transmission parameters namely, the slope of the Phillips curve and, the IRG semi-elasticity of the output gap are in line with the estimates obtained in close models for the European Union (see Table 1) 18. Furthermore, the signi cativity of both the slope of the Phillips curve and the IRG semi-elasticity of the output gap compares broadly with those of Laubach and Williams 19. As a rule, except for the standard deviations, the parameter estimates are little a ected by the choice of the ratios. Increasing 1 tends to deteriorate the signi cativity of, while only slightly diminishing this of. Turning to the second ratio, an increase in 2 pulls down the signi cativity of r which is key for our estimate of the natural rate of interest. Moreover, the larger is 2, the wider the con dence interval around the natural rate of interest grows but at the same time the greater is the log-likelihood. Finally, Figures 11 and 12 suggest that the estimates of the state variables are faintly sensitive to the ratios 20. For all this, our preferred ratios are the following: to be particularly robust to their choice. 17 For a given level of 1 taken in the range of accepted values according to the LM test, a likelihood-ratio test rejects the null-hypothesis corresponding to a given level of 2 when this last ratio is below a certain threshold. For 1 equal to 0.5, this threshold for 2 is 7 with a probability of error of 10 %. Hence, the likelihood-ratio test does not help to discriminate further between the values of 2 reported in Table The e ect of the interest rate gap on the output gap is indeed twice larger than since we consider two lags of the real rate gap in the IS curve with this same coe cient. 19 The p-values associated with the Student T for our parameters and are 5 % and 13 % respectively. In their baseline model for the United States, Laubach and Williams get p-values of 10 % and 0 % for the same parameters. 20 Larger swings are observable in the case 2 = 1. However, as previously said, this value corresponds to the case where the trend growth rate and the NRI do not have any common stochastic trend, which is unsatisfying from an economic point of view. 16

19 Table.2 Parameter estimates 1 = 0:5 1 = 0:5 1 = 0:5 1 = p 0:1 1 = 1:5 1 = 0:5 2 = 12 2 = 16 2 = 20 2 = 16 2 = 16 2 = 1 LF 200:61 200:34 200:16 200:35 200:23 199:58 avg SE 0:97 1:12 1:24 1:08 1:41 2:81 1 0:47 (4:8) 2 0:27 (2:6) 3 0:26 (2:6) 0:19 (1:8) 0:97 (13:7) 0:79 (4:8) 0:06 ( 1:5) z 0:37 (8:2) 0:89 (9:1) y 0:18 (8:2) y 0:52 (7:6) y 0:07 (1:7) r 3:12 (2:4) r 0:87 (1:7) 0:46 (4:8) 0:27 (2:6) 0:27 (2:7) 0:19 (1:9) 0:97 (13:7) 0:80 (4:9) 0:06 ( 1:5) 0:37 (8:2) 0:90 (9:5) 0:18 (8:2) 0:52 (8:4) 0:06 (1:7) 3:10 (2:2) 0:98 (1:7) 0:46 (4:7) 0:27 (2:6) 0:27 (2:7) 0:19 (2:0) 0:97 (13:7) 0:80 (5:1) 0:06 ( 1:5) 0:37 (8:2) 0:90 (9:7) 0:18 (8:2) 0:52 (9:2) 0:05 (1:6) 3:07 (2:0) 1:07 (1:6) 0:47 (4:8) 0:27 (2:6) 0:26 (2:7) 0:18 (1:9) 0:97 (13:8) 0:80 (5:1) 0:06 ( 1:5) 0:39 (7:9) 0:90 (9:6) 0:12 (7:9) 0:52 (8:6) 0:06 (1:7) 3:09 (2:2) 0:43 (4:4) 0:28 (2:7) 0:29 (2:9) 0:30 (1:8) 0:95 (13:5) 0:76 (4:1) 0:05 ( 1:2) 0:24 (9:9) 0:89 (8:1) 0:36 (9:9) 0:53 (7:3) 0:07 (1:6) 3:14 (2:3) 0:44 (4:5) 0:28 (2:7) 0:28 (2:8) 0:12 (1:7) 0:97 (13:6) 0:85 (4:8) 0:04 ( 1:1) 0:37 (7:8) 0:90 (6:5) 0:18 (7:8) 0:53 (19:7) Table 2: LF: likelihood Function - t-students in parenthesis - avg SE: average of the estimated standard 0:98 (1:7) error around the estimate of the natural rate of interest ( lter uncertainty) 1:13 (1:6) 0 3:40 (1:8) 2:28 (1:0) 17

20 Figure 3: Natural rate of interest ( 1 = 0:5; 2 = 16). 1 = y z = 0:5 and 2 = r y = 16 In order to detect signs of potential parameter nonconstancy over the sample period, we use recursive estimations. The model parameters are updated at each new observation in time, estimation sample being extended from 1979Q1-1994Q1 to 1979Q1-2002Q4. Figure 13 reports recursive estimates of and with their 80% con dence interval: the estimated parameters then appear to be fairly stable. The other model parameters are not largely a ected by these changes in the sample length either. Figure 3 plots our estimated smoothed natural rate of interest, together with the actual real rate of interest and the 90% con dence interval around the estimates of state variables. The estimated real interest rate gap o ers a valuable insight into the monetary policy stance over the last two decades. Indeed, a positive interest rate gap means that monetary policy aims at dampening the current rate of in ation. Conversely, a negative gap means that the level of the central bank s key rate gives leeway to a rise in in ation. For convenience s sake, we describe here both situations in terms of monetary policy being either tight or loose. However, a more precise terminology would refer to a disin ationary versus an in ationary policy stance. The point at stake is that the real interest rate gap is not conceptually equivalent to the di erential between the (policy driven) real interest rate and the short term real rate that a standard Taylor rule would prescribe. While the interest rate prescription of the Taylor rule aims 18

21 Figure 4: Output gap ( 1 = 0:5; 2 = 16). Figure 5: Potential output growth ( 1 = 0:5; 2 = 16). 19

22 at anchoring in ation at a given level (the in ation target of the central bank), equating the current real rate of interest with its natural counterpart only means that one has an objective of in ation stabilisation, but nothing is said about the nominal anchor. This being said, according to our measure of the real interest rate gap and taking into account lter-uncertainty, monetary policy in the euro area appears to have been signi cantly tight over three particular episodes: in the early 1980s in parallel with the Volcker era in the United States, in 1986 and from the EMS crisis of summer 1992 until Conversely, two to three episodes of signi cantly loose monetary policy are identi ed, namely in the late 1970s during the great in ation and before the vigorous tightening of the early 1980s, possibly in 1988 while the output gap of the area was rapidly reverting, and nally in 1999, mainly as a consequence of the 50 bp cut in the ECB s repo rate in April. From 2000 on, the actual real short term rate of interest appears by contrast to be fairly in line with its estimated natural counterpart, which suggests that the monetary policy stance in the euro area has been broadly appropriate since then in terms of in ation stabilisation. Turning to the output gap, Figure 4 highlights periods of excess demand around 1980, 1990 and 2000 and periods of excess supply in the mid-1980s and mid-1990s. Resulting peaks and troughs are in line with available evidence about the business cycle in the main European countries over the last two decades. The a t component satisfyingly tracks the low-frequency uctuations of potential output growth (see Figure 5) and can therefore be interpreted as the trend growth rate speci ed in Laubach and Williams (2001) once multiplied by y. According to our results, potential output growth would have reached a maximum of 3.2 % in 1989 and a minimum of 1.6% in 1982 and This nal low value of the trend growth rate partly accounts for the positive output gap at the end of the sample. Indications of such a recent decrease in the trend growth rate for the euro area are in turn consistent with empirical evidence of a slowdown in trend productivity growth in European countries in the 1990s (Maury and Pluyaud, 2004), together with the postulated end of the catching up process of American productivity levels in the mid-1990s. A proxy for the real-time estimate of the NRI is the ltered value yielded by the Kalman lter, which uses information available up to time t only (instead of T for the smoothed value) 21. Figure 6 shows that the di erences between the ltered and smoothed series of the NRI are relatively small. Both resulting interest rate gaps present roughly the same sign throughout the period. Nevertheless, the gap derived from the ltered series tends to change signs after its smoothed equivalent, which is in any case not surprising considering the information advantage of the smoothed series. Finally, di erent ltering techniques are compared in Figure 7. Two univariate lters have been used: the Hodrick-Prescott (HP) lter (1997) and the Band-Pass (BP) lter (see Baxter and King, 1999). As 21 As previously stated, this estimate is not rigorously available in real-time because it relies on estimated values for the parameters of the state-space model, which are computed on the basis of the whole sample information. 20

23 Figure 6: Filtered (one-sided) and smoothed (two-sided) natural rate of interest. regards the HP lter, two smoothness parameters are considered: 1600 and Following Staiger, Stock and Watson (2002), as well as Laubach and Williams (2003), the BP lter is used to discard the cyclical component from the real rate of interest, i.e. the frequencies corresponding to periods of up to 15 years. Consistently with their two-sided moving average representations ( nite in the case of the Baxter and King BP lter, in nite in the case of the HP lter), the two univariate lters simply track the trend of the real rate of interest, while our estimate also takes into account the actual uctuations in in ation and the level of output. More precisely, as Figure 8 shows, a positive sign of our real rate gap is contemporaneous with periods of marked disin ation while a signi cant negative sign of our real rate gap entails a rise in in ation over the same period. Besides, a persistent slowdown of the trend growth rate results as expected in a decrease in the natural rate of interest. To comment further on the relationship between the interest rate gap and in ation in the euro area, it is convenient to decompose the changes in our estimated real rate gap into three components: 1/ changes in the nominal interest rate, 2/ changes in one-quarter-ahead in ation expectations and 3/ changes in the NRI itself. Figure 9 displays this breakdown together with the corresponding interest rate gap. According 22 This last value smoothes the data slightly more than the commonly-used 1600 value. Bouthevillain et al. (2001) show that this value entails signi cant bene ts in terms of less leakage e ects (which is a ltering error corresponding to the overestimation of the variability of the cyclical component) compared to the costs related to the increase in compression e ects (which is the alternative ltering error ). 21